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2. Required the cube root of 847. Index 3)2.927883=Log. of the given number.
3. Required the square root of .093. Index 2)—2.968483=Log. of .093. —1,484241.=Quot.-Logof.304959-Ans. 4. Required the cube root of 12345. Index 3)4.091491=Log. of 12345. 1.363830=Quot-Log. of 23.116.-Ans.
5. To find the cube root of .00048. Power, or index 3)4.6812412=Log. of the number.
Root.07829735.....2.8937471=Log of the root.
Here, the divisor 3 not being exactly contain
ed in 4, augment it by 2, to make it become 6, in
which the divisor is contained just 2 times; and the 2 borrowed being prefixed to the other figures makes 2.6812412, which divided by 3 gives .8937471, therefore 2.8937471 is the Log. of the root.
6. To find the 4th root of .967845, by Logarithms. Ans. .9918624 7. To find the cube root of 2.987635. - - - Ans. 1.440265 8. To find the cube root of Rio Ans. .6827842 9. To find the value of (.001234)}. * - Ans. .00115047 10. To find the 10th root of 2.
1. GeoMETRY is that science wherein we consi. der the properties of magnitude.
2. A point is that which has no parts, being of itself indivisible; as A.
3. A line has length but no breadth; as AB figures 1 and 2.
4. The extremities of a line are points, as the extremities of the line AB are the points A and B. figures 1 and 2.
5. A right line is the shortest that can be drawn between any two points, as the line A.B. fig. 1. but if it be not the shortest, it is then called a curve line, as A.B. fig. 2.
6. A superficies or surface is considered only as having length and breadth, without thickness, as ABCD. fig. 3.
7. The extremities of a superficies are lines.
8. The inclination of two lines meeting one another (provided they do not make one continued . or the opening between them, is called an angle. Thus in fig. 4. the inclination of the line
AB to the line BC meeting each other in the point B, or the opening of the two lines BA and BC, is called an angle, as ABC.
Note, When an angle is expressed by three letters, the middle one is that at the angular point. . .
9. When the lines that form the angle are right ones, it is then called a right-lined angle, as ABC, fig. 4. . If one of them be right and the other curved, it is called a mixed angle, as B. fig. 5. If both of them be curved, it is called a curved-lined or spherical angle, as C. fig. 6.
10. If a right line, CD (fig. 7.) fall upon another right line, AB, so as to incline to neither side, but make the angles ADC, CDB on each side equal to each other, then those angles are called right angles, and the line CD a perpendicular. -
11. An obtuse angle is that which is wider or greater than a right one, as the angle ADE. fig. 7. and an acute angle is less than a right one, as EDB. fig. 7.
12. Acute and obtuse angles in general are called oblique angles.
13. If a right line CB. (fig. 8.) be fastened at the end C, and the other end B, be carried quite round, then the space comprehended is called a circle; and the curve line described by the point B, is called the circumference or the periphery of the circle ; the fixed point C, is called its centre.
14. The describing line CB. (fig. 8) is called the semidiameter or radius, so is any line from the centre to the circumference; whence all radii of the same or of equal circles are equal.
15. The diameter of a circle isa rightline drawn thro’ the centre, and terminating in opposite points of the circumference: and it divides the circle and circumference into two equal parts, called semicircles; and is double the radius, as AB or D.E. fig. 8.
16. The circumference of every circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, and these into thirds, fourths, &c. these parts being greater or less as the radius is.
17. A chord is a right line drawn from one end of an arc or arch (that is, any part of the circumference of a circle) to the other; and is the measure of the arc. Thus the right line HG, is the measure of the arc HBG. fig. 8.
18. The segment of a circle is any part thereof, which is cut off by a chord : thus the space which is comprehended between the chord HG and the arc HBG, or that which is comprehended between the said chord HG and the arc HDAEG are called segments. Whence it is plain, fig. 8.
1. That any chord will divide the circle into two segments.
2. The less the chord is, the more unequal are the segments.
3. When the chord is greatest it becomes a diameter, and then the segments are equal; and each segment is a semicircle.*
19: A sector of a circle is a part thereof less
.." the demonstration of this consult Prop. 15, Book HI. Simpson's Euclid.
than a semicircle, which is contained between two radii and an arc: thus the space contained between the two radii CH, CB, and the arc HB is a sector, fig. 8.
20. The right sine of an arc, is a perpendicular line let fall from one end thereof, to a diameter drawn to the other end: thus HL is the right sine of the arc HB.
The sines on the same diameter increase till they come to the centre, and so become the radius; hence it is plain that the radius CD is the greatest possible sine, and thence is called the whole sine.
Since the whole sine CD (fig. 8) must be perpendicular to the diameter (by def. 20.) therefore producing DC to E, the two diameters AB and DE cross one another at rightangles, and thus the periphery is divided into four equal parts, as BD, DA, AE and EB ; (by def. 10.) and so BD becomes a quadrant or the fourth part of the periphery; therefore the radius DC is always the sine of a quadrant, or of the fourth part of the circle B.D. .
Sines are said to be of as many degrees as the arc contains parts of 360: so the radius being the sine of a quadrant, becomes the sine of 90 degrees, or the fourth part of the circle, which is 360 degrees.
21. The versed sine of an arc is that part of the diameter that lies between the right sine and the circumference: thus LB is the versed sine of the arc HB. fig. 8.
22. The tangent of an arc is a right line touching the periphery, being perpendicular to the end